We define a non-Hermitian non-Abelian Berry connection or a non-Hermitian
Berry-Wilczek-Zee (BWZ) connection by[1]}
\(A_{\alpha \beta }^{\text{RL}}\left( \theta \right) =\left\langle \psi _{\alpha }^{\text{R}}\right|\partial _{\theta }\left|\psi _{\beta }^{\text{L}}\right\rangle , \) 
